arX

iv:1

909.

0502

2v2

[gr

-qc]

20

Oct

201

9

Jeans instability and turbulent gravitational collapse of Bose-Einstein Condensate

dark matter halos

Tiberiu Harko1, 2, ∗

1Department of Physics, Babes-Bolyai University,

Kogalniceanu Street, Cluj-Napoca 400084, Romania,2School of Physics, Sun Yat-Sen University, Xingang Road, Guangzhou 510275, P. R. China,

(Dated: October 22, 2019)

We consider the Jeans instability and the gravitational collapse of the rotating Bose-EinsteinCondensate dark matter halos, described by the zero temperature non-relativistic Gross-Pitaevskiiequation, with repulsive inter-particle interactions. In the Madelung representation of the wavefunction, the dynamical evolution of the galactic halos is described by the continuity and the hydro-dynamic Euler equations, with the condensed dark matter satisfying a polytropic equation of statewith index n = 1. By considering small perturbations of the quantum hydrodynamical equationswe obtain the dispersion relation and the Jeans wave number, which includes the effects of the vor-tices (turbulence), of the quantum pressure and of the quantum potential, respectively. The criticalscales above which condensate dark matter collapses (the Jeans radius and mass) are discussed indetail. We also investigate the collapse/expansion of rotating condensed dark matter halos, and wefind a family of exact semi-analytical solutions of the hydrodynamic evolution equations, derived byusing the method of separation of variables. An approximate first order solution of the fluid flowequations is also obtained. The radial coordinate dependent mass, density and velocity profiles ofthe collapsing/expanding condensate dark matter halos are obtained by using numerical methods.

PACS numbers: 04.50.+h, 04.20.Jb, 04.20.Cv, 95.35.+d

Contents

I. Introduction 1

II. Gravitationally confined Bose-Einsteindark matter halos 6

III. The Jeans instability for turbulentBose-Einstein Condensate dark matterhalos 8A. Jeans stability of the nonrotating

Bose-Einstein Condensate matter halos 9B. The effects of the rotation 11C. The Thomas-Fermi approximation 11

IV. Time evolution of Bose-EinsteinCondensate dark matter halos 12A. The stationary solution 13B. Collapsing Bose-Einstein Condensate dark

matter halos 141. The evolution equations 15

C. The first order approximation 16D. Exact numerical profiles 17

V. Discussions and final remarks 18

References 21

A. No self-similar contraction/expansion ofBose-Einstein Condensate dark matter

∗Electronic address: t.harko@ucl.ac.uk

halos 24

I. INTRODUCTION

After almost one hundred years of intensive study andresearch the properties of dark matter remain elusive.The existence of dark matter is one of the fundamen-tal assumptions of modern cosmology and astrophysics[1–7], and its nature is one of the most important openquestions in physics. Presently, all available informationon the dark sector is obtained from the study of its grav-itational interactions with astrophysical systems.Perhaps the strongest evidence for the existence of the

dark matter in the Universe comes from the study ofthe galactic rotation curves [8]. For hydrogen clouds instable circular orbits moving around the galactic cen-ter the rotational velocities first increase near the cen-ter of the galaxy, thus following the standard (New-tonian) gravitational theory, but then they remain ap-proximately constant at an asymptotic value of the or-der of vtg∞ ∼ 200 − 300 km/s. Hence the rotationcurves implies the existence of a mass profile of the formM(r) = rv2tg∞/G, where G is the gravitational constant.This fundamental result implies that within a distancer from the center of the galaxy, even at large distanceswhere very little baryonic (luminous) matter can be de-tected, the mass profile increases linearly with r. Thistype of behavior can be explained by assuming the pres-ence of a new (and exotic) mass component, interact-ing only gravitationally with ordinary matter, and whichmost likely consists of new particle(s) not (yet) includedin the standard model of particle physics. The behav-

2

ior of the galactic rotation curves still provides the mostconvincing and compelling evidence for the existence ofdark matter [9–12].

The second important astrophysical evidence for darkmatter comes from the virial mass discrepancy in galaxyclusters [13]. Galaxy clusters are giant astrophysical sys-tems formed of thousands of galaxies each, bounded to-gether by their own gravitational interaction. The galax-ies give around 1% of the mass of the clusters, while thehigh temperature intracluster gas represents around 9%of the cluster mass. But the total masses obtained bymeasuring the velocity dispersions of the galaxies exceedthe total masses of all stars in the cluster by factors ofthe order of 200 - 400 [14]. Hence, in order to explain thecluster dynamics one needs to assume the presence ofdark matter, representing around 90% of the mass of thecluster. Another strong evidence for the presence of darkmatter follows from the measurement of the temperatureof the intracluster medium, since a supplementary masscomponent is required to explain the determined depthof the gravitational potential of the clusters [14].

Several other astrophysical and cosmological observa-tions have also provided compelling evidence for the pres-ence of dark matter. From the cosmological perspective,the recent Planck satellite measurements of the CosmicMicrowave Background Radiation [15] led to the precisedetermination of the cosmological parameters. These re-sults have indicated that baryonic matter only cannot ex-plain the cosmological dynamics, and that the standardΛ Cold Dark Matter (ΛCDM) cosmological paradigm re-quiring the existence of dark matter is strongly favoredby observations. For a consistent interpretation of thegravitational lensing data the existence of dark matter isalso required [16–18].

Powerful observational evidences for the existence ofdark matter are yielded by the observations of a galacticcluster called the Bullet Cluster, consisting of two collid-ing clusters of galaxies. Due to collision of its two compo-nents that occurred in the past, in the Bullet Cluster clus-ter the baryonic and the dark matter components are sep-arated [19]. Determinations of the cosmological parame-ters from the Planck data on the cosmic microwave back-ground radiation did show convincingly that the Universeconsists of 74% dark energy, 22% non-baryonic dark mat-ter and only 4% baryonic matter [15].

Depending on the energy of the particles composingit, dark matter models can be classified generally intothree major types, cold, warm and hot dark matter mod-els, respectively, [20]. For the dark matter particle(s)the main candidates are the WIMPs (Weakly Interact-ing Massive Particles) and the axions, respectively [20].WIMPs are hypothetical (and yet undetected) heavy par-ticles, interacting through the weak force [21, 22]. Otherpopular dark matter candidates, the axions, are bosonsthat were first proposed as a solution of the strong CP(Charge+Parity) problem, which requires to explain whyquantum chromodynamics does not break the CP sym-metry [23, 24].

There are also other approaches that attempt to in-terpret the observational data without resorting to darkmatter. These explanations assume that on galactic orextra-galactic scales the law of gravity (Newtonian orgeneral relativistic) is modified. The earliest attempt toexplain the rotation curves by modified gravity is MOND(Modified Newtonian Dynamics) theory, proposed ini-tially in [25]. Other modified theories of gravity havealso been used widely as alternative explanations to darkmatter phenomenology [26–39]. For a recent review ofthe dark matter problem in some modified theories ofgravity see [40].

An attractive possibility for the detection of the pres-ence of dark matter may be provided by its possible anni-hilation into ordinary particles. If such physical processesreally take place, then a large number of positrons andgamma ray photons are produced, thus giving some clearobservational signatures for the presence of dark matter.This possibility may be supported by the detection of ex-cess positron emission in our galaxy [41–48]. Hence theexcess gamma-ray and positron emissions in our galaxycould be interpreted as coming from the annihilation ofdark matter with mass in the range of m ∼ 10−100 GeV[41–43]. For an in depth discussion of this problem, aswell as of the alternative possibilities for the interpreta-tion of the observational data see [44–48].

Dark matter models can generally give good expla-nations of the phenomenological (and unexpected) be-haviors of particle dynamics at the galactic and extra-galactic level, including the constancy of the rotationcurves, and the virial mass discrepancy. However, crucialconflicts do appear when one compares the results of thenumerical simulations of the theoretical models with theobservations. The observational data on nearly all ob-served galactic rotation curves indicate that in the pres-ence of a single pressureless dark matter component theyincrease less sharply as compared to the predictions ofthe cosmological simulations of structure formation in thestandard ΛCDMmodel. Moreover, the numerical simula-tions display dark matter density profiles that behave asρ ∼ 1/r (a cusp) at the galactic center [49]. On the otherhand observations of the galactic rotation curves showthe existence of constant density cores [50, 51]. In darkmatter physics this contradiction between theory andobservations represents the so-called core-cusp problem.Dark matter models must also face the ”too big to fail”question [52, 53]. The Aquarius simulations did showthat in the dark matter halos predicted in the standardΛCDM model the most massive subhalos are incompati-ble with the observations of the dynamics of the brightestdwarf spheroidal galaxies of the Milky Way [53]. For thedwarf spheroidal galaxies the best-fitting hosts have max-imum velocities in the range 12 km/s < Vmax < 25 km/s,while all the ΛCDM simulations give at least ten subha-los with velocities Vmax > 25 km/s. In the frameworkof the ΛCDM-based models of the satellite populationof the Milky Way these observational results cannot beinterpreted. The main contradiction between theory and

3

observations is related to the predictions of the densi-ties of the satellites, with the predicted dwarf spheroidalshaving dark matter halos more massive by a factor of ∼5than shown by the observations.

The problems mentioned above, related to the phys-ical properties of the dark matter, may be explainedif one extends the standard ΛCDM model by assumingthat dark matter particles may have some kind of self-interaction. This model, called the Self-Interacting DarkMatter (SIDM) paradigm assumes the existence of sup-plementary interactions in the dark sector [54–57] thatmay allow momentum and energy exchange between par-ticles that compose the dark matter halos. In these mod-els the basic quantity describing the dark matter haloproperties is the self-interaction cross section σDM di-vided by the dark matter particle mass m, σDM/m. Ifthe dark matter self-interactions have cross sections permass of the same order of magnitude as the strong nu-clear force, σDM/m ∼ 1g/cm−2, this would thermalizethe inner regions of dark matter halos where the bary-onic matter is concentrated. On the other hand forσDM/m ≥ 1g/cm−2 on galactic scales Self-InteractingDark Matter models can explain the uniformity as wellas the diversity of galaxy rotation curves [58–60]. Galaxyclusters also show the same diversity of properties of theirSelf-Interacting Dark Matter halos [61].

The possibility of a complex self-interaction of darkmatter particles received some observational backingfrom the study of the data obtained from the study of72 galaxy cluster collisions. The observations did includeboth ‘major’ and ‘minor’ mergers, and they were done byusing the Hubble and Chandra Space Telescopes [62]. Im-portant constraints on the non-gravitational forces act-ing on dark matter can be obtained from the study ofthe collisions between galaxy. The analysis presented in[62] gave an upper limit of σDM/m as σDM/m < 0.47cm2/g (at 95% Confidence Level). In [63] an upperlimit on the self-interaction cross-section of dark matterof σDM/m < 1.28 cm2/g (68% Confidence Level), wasobtained. Different self-interacting dark matter mod-els were investigated in [64–67]. In [68] the implica-tions of the self-interaction of dark matter for the tidalstripping and evaporation of satellite galaxies in a MilkyWay type galaxy were considered. The response of self-interacting dark matter halos to the growth of galaxypotentials using numerical simulations was investigatedin [69], and a greater diversity of dark matter halo pro-files was found. A self-interacting dark matter halo withσDM/m = 0.1 cm2/g gives a good fit to the measureddark matter density profile of A2667. The same halosimulated with σDM/m = 0.5 cm2/g does not producea core profile dense enough to fit the observational dataof A2667. Together with the previous findings in [62],these limits point towards the result that the constraintσDM/m ≥ 0.1 cm2/g is strongly disfavored for dark mat-ter collision velocities greater than 1500 km/s.

Therefore, as suggested by the above observational re-sults, we cannot reject a priori the possibility that dark

matter is a self-interacting constituent of the Universe.Physical models of dark matter in which the fundamen-tal particles are self-interacting may provide a better the-oretical explanation of the observed phenomenology atgalactic and extra-galactic scales. But from both a phe-nomenological perspective, and from a fundamental the-oretical and physical point of view, the best motivatedself-interacting dark matter models can be constructedby assuming that presently dark matter is in the form ofa Bose-Einstein Condensate.In the 1920s, by using statistical physics methods,

Bose and Einstein [70–72] proposed that at temperaturessmaller than a critical one, all integer spin particles willcondense into the lowest quantum state. In this phase mi-croscopic quantum phenomena become apparent at themacroscopic level. The Bose-Einstein Condensation pro-cess occurs when the particles in the gas become corre-lated quantum mechanically, that is, when the de Brogliethermal wavelength turns to be greater than the meaninterparticle distance. The transition to the condensatephase of the boson gas is initiated when the temperatureT is smaller than the critical one, Tcr, which is given bythe expression [73–76]

Tcr =2π~2ρ

2/3cr

ζ2/3(3/2)m5/3kB, (1)

where ρcr is the critical transition density, m is the massof the particle in the boson gas, kB is Boltzmann’s con-stant, and ζ denotes the Riemann zeta function, respec-tively.The experimental creation of Bose-Einstein Conden-

sates was first realized in dilute alkali gases in 1995 bycooling a dilute vapor of approximately two thousandrubidium-87 atoms to below 170 nK, using a combinationof laser cooling and magnetic evaporative cooling [77–79].The presence of a Bose-Einstein Condensate in a bosonicsystem is indicated, from a physical and experimentalpoint of view, by the appearance in both coordinate andmomentum space distributions of the particles of sharppeaks.Presently, the only evidence for the existence of Bose

Einstein Condensates on a microphysical scale appearedin laboratory experiments, which involve a very smallscale. On the other hand, the possibility of the exis-tence of some forms of bosonic condensates in the cosmicenvironment cannot be excluded a priori. Due to theirsuperfluid properties, in high density general relativisticobjects, like, for example, neutron or quark stars, theneutrons or the quarks could form Cooper pairs, which,once the temperature or density reach their critical val-ues, would eventually condense. Bose-Einstein Conden-sate stars may have maximum central densities of theorder of 0.1 − 0.3 × 1016 g/cm3, minimum radii in therange of 10-20 km, and maximum masses of the order of2M⊙, respectively. The study of their interesting phys-ical and astrophysical properties is presently an activefield of research [80–90].

4

Since dark matter is assumed to be a gas of bosonicparticles, it naturally follows that it may have experi-enced a phase transition during its cosmological history,and presently it is in the form of a Bose-Einstein Con-densate. The possibility that dark matter may be in theform of a Bose-Einstein Condensate was initially sug-gested in [91], and then rediscovered/reinvestigated, in[92–102]. The systematic investigation of the physicaland astrophysical properties of the Bose-Einstein Con-densate dark matter halos, based on the non-relativisticGross-Pitaevskii equation [103, 104] in the presence ofan external confining gravitational potential, was startedin [105]. A significant simplification of the mathematicaland physical formalism required for the description ofthe gravitationally bounded Bose-Einstein Condensatescan be obtained by introducing the Madelung represen-tation of the wave function. The Madelung represen-tation provides an equivalent description of the Gross-Pitaevskii equation in a form similar to the hydrody-namic fluid flow equations in classical mechanics, thatis, a continuity equation, and an Euler type equation,respectively. With the help of the Madelung hydrody-namical representation one obtains the fundamental re-sult that Bose-Einstein Condensed dark matter can bedescribed as a non-relativistic, Newtonian gas, in a grav-itational trapping potential, with the pressure and den-sity obeying a polytropic equation of state, with poly-tropic index n = 1 [106]. From this result it follows thatthe radius R of the zero temperature Bose-Einstein Con-densate dark matter halo is given by R = π

√

~2a/Gm3,where a is the scattering length [105]. The total massM of the condensate dark matter halo is obtained asM = 4π2

(

~2a/Gm3

)3/2ρc = 4R3ρc/π, where ρc is the

central density of the galactic halo. The mass of the darkmatter particle satisfies a particle mass-galactic radiusrelation given by [105]

m =

(

π2~2a

GR2

)1/3

≈ 6.73× 10−2 × [a (fm)]1/3 ×

[R (kpc)]−2/3

eV. (2)

The effectiveness of the Bose-Einstein Condensate darkmatter model was checked through the fitting of the New-tonian and general relativistic tangential velocity equa-tion to a sample of observed rotation curves data of dwarfand low surface brightness galaxies, respectively.The condensate dark matter halo may not necessarily

be at zero temperature. In [107] the thermal correctionto the dark matter density profile where obtained. Animportant result on the Bose-Einstein Condensate darkmatter halos is that their density profiles generally indi-cate the presence of an enlarged core, whose presence isdue to the strong interaction between dark matter parti-cles [108]. The investigations of the properties of the Bose- Einstein Condensate dark matter on cosmological andastrophysical scales is a very interesting and active field ofresearch [109–155]. Properties of the Fuzzy Dark Matter,a particular theoretical form of dark matter, assumed to

be formed of an extremely light boson (m ∼ 10−22 eV),with a de Broglie wavelength of the order of λ ∼ 1 kpc,were investigated in [156].

While the static properties of the Bose-Einstein Con-densate dark matter halos have been extensively investi-gated, their rotational characteristics have received lessattention. The presence of vortices in a self gravitat-ing Bose-Einstein Condensate dark matter halo, consist-ing of ultra-low mass scalar bosons, was investigated in[99]. Rotation of the dark matter may induce a harmonictrap potential for vortices. In [112] a detailed study ofthe vortices in rotating Bose-Einstein Condensate darkmatter halos was performed, and strong bounds for theshape and quantity of vortices in the halo, for interactionstrength, for the critical rotational velocity for the nucle-ation of vortices, and for the boson mass were found.In [113], by assuming that a vortex lattice forms, theeffects of rotation on a superfluid Bose-Einstein Conden-sate dark matter halo were investigated. On the rota-tion curves sub-structures similar to some observationsin spiral galaxies may form. The equilibrium proper-ties of self-gravitating, rotating Bose-Einstein Conden-sate haloes, which satisfy the Gross-Pitaevskii-Poissonequations were studied in [124]. For a wide range of theBose-Einstein Condensate dark matter physical parame-ters vortices are generated. On the other hand, vorticescannot appear for a vanishing self-interaction, and theyform when the self-interaction between dark matter par-ticles is strong enough.

One of the fundamental concepts in modern astro-physics and cosmology is that of gravitational insta-bility, initially discussed by Jeans [157]. Its impor-tance is related to the prospect of estimating the scaleof the condensations that may occur in an extendedgaseous medium under the influence of small perturba-tions. The possibility of such an estimation will pro-vide at least qualitative information regarding the for-mation of stars and of galaxies from an original cosmicmedium. The main result of the original analysis byJeans was that a self-gravitating infinite uniform gas atrest should be unstable against small perturbations pro-

portional to exp[

i(

~k · ~r − ωt)]

. The lineariazation of

the equations of the ideal hydrodynamics and the Pois-son equation results in the well-known dispersion rela-

tion ω2 = c2sk2 − 4πGρ, where cs = (γkBT/m)

1/2is the

adiabatic sound velocity, ρ is the density, T is the gastemperature, and γ = 5/3 is the ratio of specifics heats,respectively. When ω2 becomes negative, an instabilityarises once the perturbation wavelength λ = 2π/k ex-

ceeds the critical value λJ = cs√

π/Gρ, λ > λJ . Thus,an originally uniform gas, due to the instability, shouldbreak into massive components with characteristic size ofthe order of λJ [13]. The effects of the rotation and ofother physical effects on the stability of a self-gravitatingmedium were considered in [158–168]. The kinetic the-ory of the Jeans instability was developed in [169]-[173],mainly using methods from plasma physics. For a de-

5

tailed discussion of the Jeans instability and its role inastrophysics see [13].

An interesting property of the Bose-Einstein Conden-sation processes is represented by the collapse and theensuing explosion of the condensates [175]. Near a Fes-hbach resonance the atomic scattering length a can bechanged, by adjusting an external magnetic field, overa large range. Once the sign of the scattering length ischanged, a repulsive condensate of 85 Rb atoms is trans-formed into an attractive one that subsequently reachesa collapsing and an exploding phase. The collapse of aBose-Einstein Condensate was investigated by using thesemi-classical Fokker-Planck equation for a gas of freebosons in [176–178], and, for a 1/rb type potential, in[179].

The mechanisms of the gravitational collapse of theBose-Einstein Condensate dark matter halos was studiedin [132], by using a variational approach, and by choosingan appropriate trial wave function. This approach allowsthe reformulation of the Gross-Pitaevskii equation withspherical symmetry as Newtons equation of motion fora particle in an effective potential, which is determinedby the zero point kinetic energy, the gravitational en-ergy, and the particles interaction energy, respectively.The velocity of the condensate is proportional to theradial distance, with a time dependent proportionalityfunction. The collapse of the condensate ends with theformation of a stable configuration, corresponding to theminimum of the effective potential. The obtained resultsdid show that the gravitational collapse of the condenseddark matter halos can lead to the formation of stable as-trophysical systems with both galactic and stellar sizes.

It is the goal of the present paper to investigate theJeans stability and the collapsing properties of the Bose-Einstein Condensate dark matter halos in the presenceof vortices, induced by the rotation of the halo. Rotationis a general feature of galactic dynamics, probably gen-erated by some physical instability processes in the earlyUniverse. To describe the Bose Einstein Condensategalactic dark matter halos as a multi-particle bosonic sys-tem we adopt an effective approach based on the Gross-Pitaevskii equation [105]. The Gross-Pitaevskii equationgives an effective mean-field description of the gravita-tionally confined dark matter halo. The mathematicalanalysis of the condensed dark matter halos is greatlysimplified by the introduction of the Madelung represen-tation of the wave function, which allows the descriptionof the dark matter in terms of the equations of the clas-sical fluid dynamics, the continuity and the Euler equa-tion, respectively. In this approach dark matter haloscan de described as fluid structures obeying a polytropicequation of state, with polytropic index n = 1. The Eu-ler evolution equation also contains the quantum force,derived from the quantum potential, and which repre-sents a purely quantum effect. The assumptions of therotation for the condensate dark matter halo leads to thenecessity of taking into account the presence of quantizedvortices. A vortex is an excitation of the bosonic system,

and hence it is a state whose energy is higher than thatof the ground-state.

In order to investigate the Jeans stability of Bose-Einstein Condensate dark matter halos we consider theperturbation of the hydrodynamic flow equations withrespect to an appropriately chosen ground state, and weperform a local linear stability analysis, which includesthe effects of the quantum pressure, of the quantum po-tential and of the rotation, as well as the self gravity ofthe system. The dispersion relation for the propagationof the instabilities in the system is obtained, and theJeans wave number for the rotating Bose-Einstein Con-densate dark matter halo is obtained. Once the Jeanswave number is known, we can immediately obtain theJeans radius and mass, which give the critical lengthand mass scales above which condensate dark matter col-lapses. In the approximation when rotation and quantumforce are ignored, the Jeans radius and mass coincide withthe radius and mass of the static condensate. The effectsof the rotation are also considered, and the Jeans ra-dius and mass are also obtained for rotation dominateddark matter halos. Similar results can be obtained inthe framework of the Thomas-Fermi approximation, byrequiring that the total energy of the system is minimal.

Once the radius and mass of the Bose-Einstein Con-densate dark matter halo exceeds the Jeans limits for thecritical stability, gravitational collapse or expansion fol-lows. In order to study the collapse or the expansion ofthe condensate we derive first a semi-analytical solutionto the spherical hydrodynamic equations governing thetime and space evolution of a Bose-Einstein Condensatedark matter halo. First we reduce the system of the cou-pled nonlinear partial equations describing the evolutionof the galactic halo to a single, strongly nonlinear par-tial differential equation for the mass M = M(r, t) of thecondensate. Then we factorize the mass function intoproducts of a time-dependent factor and another factordepending only on the spatial variable r. This factor-ization means that both spatial and temporal profiles ofthe gravitational, mechanical and thermodynamic vari-ables of the halo have a universal behavior, once they arefactorized appropriately. To study the spatial profiles ofthe collapsing/expanding rotating dark matter halos weuse approximate and numerical methods. A first orderapproximative solution of the mass function is explicitlyobtained. The general mass, density and velocity spa-tial profiles are obtained by numerically integrating themass equation. In relation with the general propertiesof the gravitational collapse we also show that homolo-gous solutions to the hydrodynamic equations describingthe time evolution of a Bose-Einstein condensate darkmatter, wherein thermodynamic variables factorize intoproducts of a time-dependent factor and another factordepending only on the scaled spatial variable ξ ≡ r/R(t),do not exist. This factorization means that spatial pro-files of the thermodynamic variables remain time invari-ant. Since Bose-Einstein Condensate dark matter halosdo not have this property, it follows that the dynamics

6

of the collapse of a condensed galactic halo is very dif-ferent from other types of gravitational collapse, like, forexample, the pre-supernova stellar core collapse.

The present paper is organized as follows. The basicproperties of the Bose-Einstein Condensate dark matterhalos are presented in Section II, where, with the use ofthe Madelung representation, the evolution equations ofthe condensed galactic halos are formulated in terms ofthe continuity and Euler equations of classical hydrody-namics. The Jeans instability of condensed dark matterhalos confined by their own gravitational field is discussedin Section III, by fully taking into account the effects ofthe rotation and the quantum effects. The limiting casesof the general dispersion relation are considered in de-tail. The time evolution of the Bose-Einstein Conden-sate dark matter halos is investigated in Section IV, andit is shown that the hydrodynamic equations describingthe dynamics of a rotating polytropic gas with polytropicindex n = 1 do admit a separable solution, in which allphysical quantities can be expressed as products of twofunctions, one depending on time only, and the second afunction of the radial coordinate only. An approximatefirst order solution of the evolution equations is also ob-tained. The spatial dependence of the physical quantitiesis investigated by using numerical methods. We discussand conclude our results in Section V. In Appendix A weexplicitly show that the hydrodynamic evolution equa-tions describing a polytropic fluid with polytropic indexn = 1 do not admit self-similar homologous solutions.

II. GRAVITATIONALLY CONFINEDBOSE-EINSTEIN DARK MATTER HALOS

Generally, Bose-Einstein condensation processes takeplace in a Bose gas with particle number density nwhen the thermal de Broglie wave length λdB =√

2π~2/mkBT , exceeds the mean inter-particle distance

n1/3. Then, as a result, the wave packets percolate inspace. The critical condensation temperature can then beobtained qualitatively as T ≤ 2π~2n2/3/mkB [105]. Un-der the assumption of an adiabatic cosmological expan-sion of the Universe, the temperature dependence of thenumber density of the particle is T ∝ n2/3. Hence Bose-Einstein condensation occurs if the mass of the particlesatisfies the condition m < 1.87 eV [180], and thereforeparticles satisfying this mass limit could Bose-Einsteincondense, and form large scale cosmological or astrophys-ical structures.

Due to the low temperature of the Bose-Einstein Con-densates, their physical properties can be understoodwithin the so called mean-field approximation. The suc-cess of the mean field approach is determined by thedilute nature of the Bose gases. We can always writethe Bose field operator as a sum of the condensate wavefunction and an operator describing the non-condensedbosons, so that Ψ (~r, t) = Ψ (~r, t) + Ψ′ (~r, t), where

Ψ (~r, t) ≡⟨

Ψ (~r, t)⟩

is the average value of Ψ (~r, t), and

Ψ′ (~r, t) represents the fluctuations in the system. Thesingle-particle density matrix ρ is given by ρ (~r, ~r ′) =⟨

Ψ+ (~r) Ψ (~r ′)⟩

, where Ψ+ (~r) is the field operator creat-

ing a particle at a point ~r, and Ψ (~r ′) is the field operatorannihilating a particle at ~r ′. In a dilute Bose gas closeto T = 0, one can neglect with a very good approxima-tion the non-condensed bosons Ψ′ (~r, t). In this case themean-field order parameter is given exactly by the quan-tum mechanical wave function Ψ (~r, t), with well definedphase. Hence, the zero temperature dynamics of a Bosegas consisting of confined weakly interacting particles isdescribed by a mean-field macroscopic wave function Ψ.The wave function of the condensate behaves like a com-plex order parameter whose absolute value and phasecontains all relevant information about the Bose-EinsteinCondensate system, and satisfies a nonlinear Schrodingerequation, called the Gross-Pitaevskii equation [103, 104].Hence, gravitationally confined Bose-Einstein Conden-

sate dark matter halos are described by the coupledGross-Pitaevskii and Poisson equations for the conden-sate wave function Ψ, with a quartic non-linear term,given by [73–76]

i~∂

∂tΨ(~r, t) = HΨ(~r, t) =

[

− ~2

2m∇2 +mVext(~r, t) +

U0|Ψ(~r, t)|2]

Ψ(~r, t), (3)

where U0 = 4π~2a/m, a is the coherent scattering length(defined as the zero-energy limit of the scattering ampli-tude fscat), m is the mass of the condensate particle, andVext is the external potential. In the following we assumethat the exterior potential Vext(~r, t) is the gravitationalpotential Vgrav, Vext(~r, t) = Vgrav (~r, t) ≡ φ (~r, t). Fora single component condensate dark matter halo φ (~r, t)satisfies the Poisson equation

∇2φ(~r, t) = 4πGm |Ψ(~r, t)|2 = 4πGρ(~r, t), (4)

where ρ = mn (~r, t) = m |Ψ(~r, t)|2 is the mass density

inside the Bose-Einstein condensate, n (~r, t) = |Ψ(~r, t)|2is the particle number density, and G is the gravita-tional constant, respectively. The probability density|Ψ(~r, t)|2 is normalized according to

∫

V n (~r, t) d3~r =∫

V |Ψ(~r, t)|2 d3~r = N , where N is the total particle num-ber in the dark matter halo, which can be obtained byintegrating the norm of the wave function over the entirevolume V of the Bose-Einstein Condensate.The time dependent Gross-Pitaevski Eq. (3) can be

derived from the action principle δ∫ t2t1

Ldt = 0, where

the Lagrangian L is given by [181]

L =

∫

i~

2

(

Ψ∗∂Ψ

∂t−Ψ

∂Ψ∗

∂t

)

dV − E, (5)

7

where E =∫

εdV is the total energy, with the energydensity ε given by

ε =~2

2m|∇Ψ|2 +mφ |Ψ|2 + u0

2|Ψ|4 . (6)

If one multiplies Eq. (3) by Ψ∗ and subtracts the com-plex conjugate of the resulting equation, one arrives atthe equation

∂ |Ψ|2∂t

+∇ ·[

~

2im(Ψ∗∇Ψ−Ψ∇Ψ∗)

]

= 0. (7)

Eq. (7) has the form of a continuity equation for theparticle density, and can be written as a continuity equa-tion,

∂ρ

∂t+∇ ·~j = 0, (8)

where the momentum ~j of the condensate is defined by

~j = ρ~v =~

2i(Ψ∗∇Ψ −Ψ∇Ψ∗) , (9)

where ~v is the condensate velocity. By representing thewave function as [73–76]

Ψ (~r, t) = ϕ (~r, t) exp [iS (~r, t)] , (10)

where S (~r, t) is the phase of the wave function, we obtain

~j = mϕ2 (~r, t)~

m∇S (~r, t) , (11)

which gives

ϕ (~r, t) =

√

ρ (~r, t)

m, (12)

and

~v =~

m∇S (~r, t) , (13)

respectively.By substituting the wave function (10) into the Gross-

Pitaevskii equation (3), it follows that it can be decom-posed into two independent equations, the continuityequation (8), and the Hamilton-Jacobi equation

− ~∂S

∂t= − ~

2

2mϕ∇2ϕ+

1

2m~v2 +mφ+ U0ϕ

2. (14)

By taking the gradient of the above equation we obtain

d~v

dt=

∂~v

∂t+ (~v · ∇)~v = −1

ρ∇p−m∇φ+

1

m∇(

~2

2m

∇2√ρ√ρ

)

. (15)

where p = u0ρ2 is the quantum pressure, with u0 =

2π~2a/m3, while the last term in Eq. (15) is the quan-tum force, derived from the quantum potential VQ =

−(

~2/2m

)

∇2√ρ/√ρ. By taking into account the math-

ematical identity

(~v · ∇)~v =1

2∇~v 2 − ~v × (∇× ~v) , (16)

we can rewrite the Euler Eq. (15) as

∂~v

∂t+

1

2∇~v 2 = −1

ρ∇p−m∇φ+

1

m∇(

~2

2m

∇2√ρ√ρ

)

+ ~v × (∇× ~v) . (17)

The integration of Eq. (8) over the volume V of thecondensate, and by using use of the Gauss theorem, leadsimmediately to the equation of the particle number con-servation, which can be formulated as

∂

∂tN +

1

m

∫

S

ρ~v · ~ndS = 0, (18)

From Eq. (18) it follows that the time variation of theparticle number can be due only to the loss or gain of thedark matter particles moving in or out of the condensatethrough the surface S that encompasses the total volumeV . For a static condensate, ~v ≡ 0, and ρ(R) ≡ 0, whereR is the radius of the condensate, a condition which im-plies that the particle flux ~j(R) = ρ(R)~v ≡ 0. Hence,from Eq. (18), it follows that the total particle numberin the Bose-Einstein Condensate dark matter halo at zerotemperature is a constant, N = constant.For superfluid for which the phase of the wave function

is nonsingular, the continuity and Euler Eqs. (8) and (17)represent a potential (irrotational) flow, since the condi-tion ∇ × ~v = 0 is always satisfied, due to the definition(13) of the fluid velocity.The vorticity can be present in a Bose-Einstein Con-

densate due to the phase singularity lines [182–184].Since the phase of the wave function is defined within afactor of 2π only, Eq. (13, defining the condensate veloc-ity, implies that the circulation of the velocity field alonga closed contour must be quantized in units of ~/m, ac-cording to the rule [182–184]

Γ =

∮

~v · d~l = n~

m, (19)

where n, called the topological charge of the flow, is aninteger number. It also represents the winding numberof the phase of the wave function along the contour. Toobtain a non-zero circulation, the contour must windsaround a line of zero density, along which the phase, aswell as the velocity field, are no longer defined. Thenucleation of the quantized vortices is a powerful ex-perimental evidence for the existence of the macroscopicwave function describing the Bose-Einstein Condensate.In laboratory experiments the observation of the quan-tized vortices is quite difficult due to the smallness of thesize of the vortex core. This size is of the order of the so

8

called healing length [75]

ξh =~

mvs, (20)

where vs is the velocity of the sound in the condensate(for He4, the healing length is of the order of a fewangstroms).In the case of the presence in the condensate of a suf-

ficiently large number of vortices we can write [182–184]

∇× ~v = 2~Ω, (21)

where ~Ω is the macroscopic angular velocity of the con-densate, given by

∣

∣

∣

~Ω∣

∣

∣ = π~

mnV , (22)

where nV is the areal vortex number density, defined asnV = NV /A⊥, and obtained by assuming that the singu-lar velocity fields of the vortices are distributed uniformlyin the plane of rotation with area A⊥. As we move awayfrom the vortex, the velocity slowly decreases. If we movetowards the vortices then the superfluid density tends tozero.This approximation is called the approximation of the

distributed vorticity, and it has been used very success-fully to describe the dynamics of Bose-Einstein Conden-sates containing vortex lattices. In the distributed vortic-ity approximation we introduce a rotational componentin the velocity field of the condensate, so that we canwrite [182–184]

~v = ~vI + ~vR, (23)

where ~vI represents the irrotational component, while

~vR = ~Ω× ~r.

III. THE JEANS INSTABILITY FORTURBULENT BOSE-EINSTEIN CONDENSATE

DARK MATTER HALOS

To study the astrophysical and cosmological implica-tions of the presence of vortex turbulence in the conden-sate Bose-Einstein dark matter, and its impact on struc-ture formation, we assume that as a result of an exter-nal/internal perturbation the Bose-Einstein Condensatedark matter structure becomes unstable, and evolves intime like a non-relativistic, dissipationless fluid. The dy-namical evolution of the condensed dark matter halo isdescribed by the continuity equation, the hydrodynami-cal Euler equation and the Poisson equation, respectively,which can be written as

∂ρ

∂t+∇ · (ρ~v) = 0, (24)

∂~v

∂t+

1

2∇~v2 = −1

ρ∇p+ ~g +

1

m∇(

~2

2m

∇2√ρ√ρ

)

+ 2~v× ~Ω,

(25)

∇× ~g = 0,∇ · ~g = −4πGρ, (26)

where ~g = −∇φ is the gravitational acceleration. Wetake as the initial (unperturbed) state of the condenseddark matter the state characterized by the absence of the”real” gravitational interactions, ~g = ~g0 = 0, of the hy-drodynamical flow, ~v = ~v0 = 0, and by constant valuesof the density and pressure, ρ = ρ0, p = p0, respectively.Moreover, we assume an initial rotation of the dark mat-

ter cloud, so that ~Ω(0) = ~Ω0 6= 0. The instability inthe dark matter halo results in the appearance of thegravitational interaction in the system, and of the smallperturbations of the hydrodynamical quantities, which inthe first order linear approximation can be representedas

ρ = ρ0 + δρ, p = p0 + δp, ~Ω = ~Ω0 + δ~Ω,

~v = ~v0 + δ~v, ~g = ~g0 + δ~g, (27)

where we assume −1 << δρ/ρ0 << 1, −1 << δp/p0 <<

1, |δ~g| / |~g0| << 1, and∣

∣

∣δ~Ω∣

∣

∣ /∣

∣

∣

~Ω0

∣

∣

∣ << 1, respec-

tively. Therefore in the first order linear approximationEqs. (25) and (26) take the form

∂δρ

∂t+ ρ0∇ · δ~v = 0, (28)

∂δ~v

∂t= −v2s

ρ0∇δρ+ δ~g +

~2

4m2ρ0∆∇δρ+ 2δ~v × ~Ω0, (29)

∇× δ~v = −2~Ω0δρ

ρ0, ∇× δ~g = 0, ∇· δ~g = −4πGδρ, (30)

where we have introduced the adiabatic speed of soundvs in the unperturbed dark matter fluid, defined as

vs =

√

δp

δρ

∣

∣

∣

∣

∣

ρ=ρ0

=

√

∂p

∂ρ

∣

∣

∣

∣

∣

ρ=ρ0

=√

2u0ρ0 =

√

4π~2a

m3ρ0 =

1.57× 107 ×( a

10−3 fm

)1/2(

ρ010−24 g/cm3

)1/2

×( m

meV

)−3/2

cm/s. (31)

To obtain the first of the equations in (30) we havetaken the variation of Eq. (28) as follows: ∇×(v0 + δ~v) =

∇ × δ~v = 2δ~Ω, δ∣

∣

∣

~Ω∣

∣

∣ = (π~/m)nV (δnV /nV ) =

−∣

∣

∣

~Ω∣

∣

∣

0δρ/ρ0, where we have assumed that δnV /nV =

δρV /ρV = −δρ/ρ0, that is, the relative variation of thecondensate areal density ρV = mnV is inversely propor-tional to the variation of the density of the condensatenormalized with respect to the initial density (the higherthe mass density of the vortices, the lower the mass den-sity of ordinary matter in the condensate).After taking the partial derivative with respect to the

time of the continuity equation (28), and applying the

9

∇ operator to Eq. (29), we obtain the propagation equa-tion of the density perturbations in the Bose-EinsteinCondensate dark matter fluid as

∂2δρ

∂t2= v2s∇2δρ+ 4πGρ0

(

1 +~Ω20

πGρ0

)

δρ− ~2

4m2∇4δρ.

(32)To obtain Eq. (32) we have also used the mathematical

identity ∇·(

δ~v × ~Ω0

)

= ~Ω0 · (∇× δ~v)− δ~v ·(

∇× ~Ω0

)

=

~Ω0 · (∇× δ~v).We look now for a solution of Eq. (32) of the form

δρ =∑

A(

ω,∣

∣

∣

~k∣

∣

∣

)

exp[

i(

ωt− ~k · ~r)]

. Hence after sub-

stitution into the equation of this representation of thedensity variation we obtain for each component ω thefollowing dispersion relation,

ω2 = v2s~k2 − 4πGρ0

(

1 +~Ω20

πGρ0

)

+~2

4m2~k4. (33)

The Jeans wave number∣

∣

∣

~k∣

∣

∣

Jcorresponds to the van-

ishing of the eigenfrequency ω, and it is given as a solu-tion of the algebraic equation

~2

4m2~k4J + v2s

~k2J − 4πGρ0

(

1 +~Ω20

πGρ0

)

= 0. (34)

Hence for ~kJ we find

~k2J =2m2

~2v2s

√

√

√

√1 +4πG~2

m2v4sρ0

(

1 +~Ω20

πGρ0

)

− 1

=

2

ξ2h

√

√

√

√1 +4πGξ2hv2s

ρ0

(

1 +~Ω20

πGρ0

)

− 1

, (35)

where we have introduced the healing length of the con-densate dark matter halo as defined by Eq. (20).From the dispersion equation, rewritten as

ω2 =(

~k2 − ~k2J

)

[

v2s +~2

4m

(

~k2 + ~k2J

)

]

, (36)

it follows that for∣

∣

∣

~k∣

∣

∣ <∣

∣

∣

~k∣

∣

∣

J, the angular frequency ω

becomes imaginary. This situation corresponds to aninstability of the Bose-Einstein Condensate dark mat-ter fluid - δρ can either exponentially increase or de-crease, thus leading to a gravitational rarefaction (orcondensation) in the dark matter halo. Hence, for∣

∣

∣

~k∣

∣

∣ <∣

∣

∣

~k∣

∣

∣

J, ω ∝ ±vs

√

~k2 − ~k2J = iImω, and therefore

δρ ∝ exp [± |Imω| t]. When the mass of the Bose-EinsteinCondensate dark matter halo is greater than the mass of

a sphere with radius RJ = 2π/∣

∣

∣

~k∣

∣

∣

J, a gravitational in-

stability occurs, and the dark matter cloud of particles

would collapse. The critical mass delimitating the two

phases is the Jeans mass MJ = (4π/3)(

2π/∣

∣

∣

~k∣

∣

∣

J

)3

ρ0.

For the Bose Einstein Condensate dark matter halos inthe presence of rotation and vortices, the Jeans radius isgiven by

RJ =

√2π~

mvs

√

√

√

√1 +4πG~2

m2v4sρ0

(

1 +~Ω20

πGρ0

)

− 1

−1/2

,

(37)while for the Jeans mass we obtain

MJ =8√2π4

~3

3m3v3s×

√

√

√

√1 +4πG~2

m2v4sρ0

(

1 +~Ω20

πGρ0

)

− 1

−3/2

ρ0.(38)

A. Jeans stability of the nonrotating Bose-EinsteinCondensate matter halos

We consider first the case of the collapse of the veryslowly rotating condensate dark matter halos, having

χ =~Ω20

πGρ0= 4.774× 10−2 ×

∣

∣

∣

~Ω∣

∣

∣

0

10−16 s−1

2(

ρ010−24 g/cm3

)−1

<< 1. (39)

Hence in this approximation the terms containing∣

∣

∣

~Ω2∣

∣

∣

can be safely neglected in the expressions of the Jeanswave number, radius and mass. Therefore for the Jeanswave number of the slowly rotating/nonrotating BoseEinstein Condensate dark matter halo we find the ex-pression

∣

∣

∣

~k∣

∣

∣

J=

√2mvs~

(√

1 +4πG~2

m2v4sρ0 − 1

)1/2

. (40)

If the condition

4πG~2

m2v4sρ0 =

Gm4

4πa2~2ρ0<< 1, (41)

or, equivalently,

ρ0 >> ρcr =Gm4

4πa2~2= 4.817× 10−66 ×

( m

meV

)4 ( a

10−3 fm

)−2

g/cm3, (42)

10

is satisfied, by power expanding the square root we obtain

∣

∣

∣

~k∣

∣

∣

J≈√

4πGρ0v2s

=

√

Gm3

~2a= 5.828× 10−23 ×

( m

meV

)3/2 ( a

10−3 fm

)−1/2

cm−1, ρ0 >> ρcr.(43)

In the opposite limit(

4πG~2/m2v4s

)

ρ0 >> 1, or

ρ0 <<Gm4

4πa2~2, (44)

we obtain

∣

∣

∣

~k∣

∣

∣

J≈(

16πGm2

~2ρ0

)1/4

= 1.759× 10−12( m

meV

)1/2

×(

ρ010−24 g/cm3

)1/4

cm−1, ρ << ρcr. (45)

The effective radius RJ of the Bose-Einstein Conden-sate dark matter configuration at the onset of the gravi-tational instability is given by

RJ =

√2π~

mvs

(√

1 +4πG~2

m2v4sρ0 − 1

)−1/2

, (46)

and it has the limiting values

RJ ≈√

π

Gρ0vs =

√

4π2~2a

Gm3= 34.96×

( m

meV

)−3/2 ( a

10−3 fm

)1/2

kpc, ρ0 >> ρcr, (47)

and

RJ ≈(

π3~2

Gm2ρ0

)1/4

= 3.57× 1012 ×( m

meV

)−1/2

×(

ρ010−24 g/cm3

)−1/4

cm, ρ0 << ρcr, (48)

respectively. The Jeans mass of the static Bose-EinsteinCondensate dark matter halo is given by

MJ =16π4

~3ρ0

3√2m3v3s

(√

1 +4πG~2

m2v4s− 1

)−3/2

. (49)

For ρ0 >> ρcr, for the Jeans mass we obtain the ap-proximate expression

MJ ≈ 32π4

3

(

~2a

Gm3

)3/2

ρ0 = 2.623× 1012 ×( a

10−3 fm

)3/2

×( m

meV

)−9/2

×(

ρ010−24 g/cm3

)

M⊙, ρ >> ρcr. (50)

For ρ0 << ρcr we find

MJ ≈ 32π4

3

(

~2

16πGm2

)3/4

ρ1/40 = 1.906× 1014 ×

( m

meV

)−3/2(

ρ010−24 g/cm3

)1/4

g, ρ0 << ρcr.(51)

The maximum mass of the high density astrophysicalobjects, like white dwarfs and neutron stars, describedby polytropic equations of state, was derived by Landauand Chandrasekhar, and it can be approximated by theChandrasekhar mass [185], [186],

MCh ≈(

~c

G

)3/21

m2B

, (52)

where by mB we have denoted the mass of the parti-cles that give the most important contribution to thetotal stellar mass (protons and neutrons in the case ofthe white dwarfs and neutron stars, respectively) [185],[186]. Hence, the maximum mass of a degenerate staris fully determined by fundamental physical constantsonly, with the possible exception of some numerical fac-tors that depend on the chemical composition of the star.

Interestingly enough, the Jeans mass for Bose-EinsteinCondensate dark matter halos given by Eq. (50) can bealso written in a form similar to the Chandrasekhar max-imum mass, if we associate to the dark matter particlesan effective mass meff defined as

meff =1√ρ0

(

cm3

~a

)3/4

, (53)

so that

MJ =8√2π4

3

(

~c

G

)3/2

m−2eff . (54)

For the adopted values of the physical parameters of thedark matter halos the numerical value of the effectivemass is of the order of meff ≈ 4.52× 10−29 g.

On the other hand, one could also ask the question ifBose-Einstein Condensate dark matter could form astro-physical objects and structures with the limiting massgiven by the Chandrasekhar mass as given by Eq. (52).For the adopted mass of the dark matter particle m =1 meV = 1.78×10−36 g, the Chandrasekhar limit gives amass of the order of MCh = 1.62×1034M⊙ = 3.25×1057

g, which exceeds by three orders of magnitude the totalmass of the ordinary matter in the Universe, 4.5× 1054 g[? ]. Therefore, it follows that the Chandrasekhar masslimit cannot lead to realistic descriptions of the physicalproperties of condensate dark matter clouds composed ofparticles having masses of the order of the mass valuesassumed for condensate dark matter particles.

11

B. The effects of the rotation

Generally, in the slow rotation limit when the factor~Ω20/πGρ0 is still small, or of the order of unity, the rota-

tion modifies the Jeans parameters (Jeans wavelength,radius and mass, respectively), by a factor of 1 + χ.Hence, if, for example, the condition

4πG~2

m2v4sρ0

(

1 +~Ω20

πGρ0

)

<< 1, (55)

holds, then we immediately obtain

∣

∣

∣

~k∣

∣

∣

J≈√

Gm3

~2a

√

1 +~Ω20

πGρ0, (56)

RJ ≈√

4π2~2a

Gm3

(

1 +~Ω20

πGρ0

)−1/2

, (57)

and

MJ ≈ 32π4

3

(

~2a

Gm3

)3/2(

1 +~Ω20

πGρ0

)−3/2

ρ0, (58)

respectively. However, with the increase of the initialrotational velocity of the Bose-Einstein condensed darkmatter halo, the effects of the rotation become more andmore important. For a rotational angular velocity of∣

∣

∣

~Ω0

∣

∣

∣ = 10−14 s−1, we obtain χ = ~Ω20/πGρ0 = 477.47,

while for∣

∣

∣

~Ω0

∣

∣

∣ = 10−12 s−1 we have χ = 4. 77 × 106.

Hence for the astrophysical and physical description ofdark matter halos rotating with such initial angular ve-locities the effect of the rotation must be taken into ac-count. Hence for rapidly rotating dark matter condensedclouds the Jeans wavelength becomes

∣

∣

∣

~kJ

∣

∣

∣≈

√2m

~vs

√

1 +4~2~Ω2

0

m2v4s− 1

1/2

, (59)

or, equivalently,

∣

∣

∣

~kJ

∣

∣

∣ ≈√

8πa

mρ0

√

1 +m4~Ω2

0

4π2~2a2ρ20− 1

1/2

. (60)

For the adopted values of the physical and astrophys-

ical quantities the parameter m4~Ω20/4π

2~2a2ρ20 has gen-

erally small values, much smaller than one. Therefore we

ca expand the square root into power series, and in thefirst approximation we obtain

∣

∣

∣

~kJ

∣

∣

∣ ≈√

2

π

m3/2

~√aρ0

∣

∣

∣

~Ω0

∣

∣

∣ = 1.801× 10−19( m

meV

)3/2

×

( a

10−3 fm

)−1/2(

ρ010−24 g/cm3

)−1/2

∣

∣

∣

~Ω0

∣

∣

∣

10−12s−1cm−1.

(61)

Hence the Jeans radius and the Jeans mass of rapidlyrotating Bose-Einstein condensate dark matter halos canbe obtained as

RJ ≈√2π3/2

~√aρ0

m3/2∣

∣

∣

~Ω0

∣

∣

∣

= 1.132× 10−2 ×( m

meV

)−3/2

×

( a

10−3 fm

)1/2(

ρ010−24 g/cm3

)1/2

∣

∣

∣

~Ω0

∣

∣

∣

10−12s−1

−1

kpc,

(62)

and

MJ ≈ 8√2π11/2

3

~3(

aρ5/30

)3

2

m9/2∣

∣

∣

~Ω0

∣

∣

∣

3 = 88.752×( m

meV

)−9/2

×

( a

10−3 fm

)3/2(

ρ010−24 g/cm3

)3/2

×

∣

∣

∣

~Ω0

∣

∣

∣

10−12s−1

−3(

ρ010−24 g/cm3

)

M⊙, (63)

respectively. Interestingly enough, in the case of the ro-tation dominated Bose-Einstein Condensate dark matterhalos, their physical and astrophysical properties are in-dependent of the gravitational constant G, and they arefully determined by the fundamental quantum parame-ters m and a, characterizing the condensate, as well as

by the angular velocity of the halo∣

∣

∣

~Ω0

∣

∣

∣.

C. The Thomas-Fermi approximation

Finally, as a last step in our general analysis of theproperties of the Bose-Einstein Condensate dark matterhalos we introduce the Thomas-Fermi approximation forthe hydrodynamical description of the condensate. Inorder to perform our analysis, we decompose the totalenergy, Etot, into kinetic, interaction, quantum and grav-itational contributions [73–76],

Etot = Ekin + Eint + Eq + Egrav, (64)

where

Ekin =~2

2m

∫

[

√

ρ (~r, t)~v (~r, t)]2

d3~r, (65)

12

Eint = U0

∫

[ρ (~r, t)]2d3~r, (66)

Eq =~2

2m

∫

[

∇√

ρ (~r, t)]2

d3~r, (67)

Egrav =

∫

ρ (~r, t)φ (~r, t) d3~r. (68)

The kinetic energy is determined by the velocity fieldof the dark matter fluid. The interaction energy is relatedto the interaction of the particles in the condensate. Thequantum energy is determined by square of the gradientof the square root of the density of the condensate, whilethe gravitational energy has its origin from the externalpotential of the field, acting as a trapping potential.In the Thomas-Fermi approximation, we neglect

in the total anergy balance the kinetic energy term−(

~2/2m

)

∆ of the condensate particles. To obtain theconditions of the validity of the Thomas-Fermi approx-imation we consider a system composed of N bosonsin a volume V , extended over a radius R, in the pres-ence of a confining gravitational potential. The to-tal mass of the system is denoted by M . Moreover,we assume that all bosons are in the same quantumstate. The total energy Etot of the system is given byE = Ekin + Eint + Egrav. For a single particle thekinetic energy can be approximated as ~

2/2mR2 [75],and hence the total kinetic energy of the system canbe obtained as Ekin = N~

2/2mR2. The interaction en-ergy is given by Eint = (1/2)

(

N2/V)

mg [75], while thegravitational potential energy can be approximated asEgrav = −αGM2/R, where α is a constant. For then = 1 polytropic equation of state α = 3/4 [106]. There-fore for the total energy of the bosonic system in a grav-itational field we obtain the expression

Etot = N~2

2mR2+

3

2N2 ~

2a

mR3− αGM2

R. (69)

If the condition

3Na

R>> 1, (70)

is satisfied, then the interaction energy is much largerthan the kinetic energy, Eint >> Ekin. In order tocheck the validity of the condition in an astrophysi-cal setting, we consider a galactic condensate of massM = 1010M⊙ = 2 × 1043 g. Then number of the darkmatter particle is of the order of N = M/m = 1.12×1079

particles. If the radius of the condensate is R = 10 kpc= 3 × 1022 cm, then the quantity 3Na/R = 1.12× 1041,a number that is obviously much greater than one,3Na/R >> 1. Hence in galactic Bose-Einstein Conden-sate dark matter halos the contribution to the total en-ergy of the kinetic energy of the particles can be safelyneglected when compared to the interaction energy of

the bosonic fluid. Therefore, in the case of bosonic sys-tems consisting of a very large number of particles, theThomas-Fermi approximation gives an excellent descrip-tion of the physical and astrophysical properties of thecondensate, corresponding to a precision level that canbe recognized as exact.On the other hand, the physical length scales of the

order of R ≈√

m/4πρa, where ρ is the mean dark mat-ter density, the Thomas-Fermi approximation is not validanymore. For ρ = 10−24 g/cm3, we obtain R ≈ 377.65cm, a length scale that is insignificant from the pointof view of the galactic dark matter distribution. Forthe gravitational energy of the Bose-Einstein Condensatedark matter halo we obtain the value Egrav = 8.89×1056

ergs, why for the considered numerical values of the pa-rameters of the dark matter halos the interaction energyhas the value Eint = 1.35× 1056 ergs. Hence in gravita-tionally trapped galactic size condensates the interactionenergy is of the same order of magnitude as the gravi-tational energy of the bosons. However, it is importantto point out that Egrav > Eint, a relation that showsthat the dark matter halo is trapped by its gravitationalenergy.Hence in the Thomas-Fermi approximation the total

energy of the condensate dark matter halo is given by

Etot =3

2M2 ~

2a

m3

1

R3− 3

4

GM2

R. (71)

The equilibrium radius corresponding to the minimumof the total energy is given by the solution of the algebraicequation

dEtot

dR= −9

2M2 ~

2a

m3

1

R4+

3

4

GM2

R2= 0, (72)

given by

R =

√

6~2a

Gm3, (73)

which, except some numerical factors, coincides withEq. (47), giving the Jeans radius of the condensate darkmatter halo.Therefore in our subsequent analysis of the gravita-

tional collapse of Bose-Einstein Condensate dark matterhalos we will neglect, with a very good approximation,the kinetic energy term in the Gross-Pitaevskii Eq. (3),and, consequently, the quantum force, derived from thequantum potential, in the hydrodynamic description ofthe dynamics of the condensate. In the Thomas-Fermiapproximation the chemical potential of the static con-densate is given by

µ = mφ (~r) + U0ρ (~r) . (74)

IV. TIME EVOLUTION OF BOSE-EINSTEINCONDENSATE DARK MATTER HALOS

We consider now a Bose-Einstein Condensate darkmatter halo falling inward from an initial rest state. The

13

halo is only under the effects of its own self-gravity. Thebasic equations describing the dynamical evolution ofthe condensate dark matter halo are then the continuityequation, the Euler equation, and the Poisson equation,which in spherical symmetry are given by

∂ρ

∂t+

1

r2∂

∂r

(

r2ρv)

= 0, (75)

∂v

∂t+ v

∂v

∂r= −2u0

∂ρ

∂r− ∂φ

∂r+

2

3Ω2r, (76)

1

r2∂

∂r

(

r2∂φ

∂r

)

= 4πGρ, (77)

where in the Euler equation we have used the polytropicequation of state p = u0ρ

2 of the condensate. The lastterm in Eq. (76) takes into account the effects of the ro-tation, and it has been obtained as follows. If θ is thecolatitude of a given point in the condensate with re-spect to the rotation axis, the centrifugal acceleration ofthe former is Ω2r sin θ. Its radial component is Ω2r sin2 θ.In order to preserve the spherical symmetry of the darkmatter halo we will neglect all the other components ofthe centrifugal acceleration, and take into account themean value of Ω2r sin2 θ on a sphere of radius r. Con-sequently we obtain for the centrifugal acceleration theexpression 2Ω2r/3 [187].The Poisson equation (77) can be immediately inte-

grated to give

∂φ

∂r=

GM (r, t)

r2, (78)

where M(r, t), the total mass of the dark matter halowithin radius r, is defined as

M(r, t) = 4π

∫ r

0

r′2ρ (r′, t) dr′. (79)

We multiply now the continuity equation (75) by 4πr2.After using the relation 4πr2ρ = ∂M/∂r, and integratingwith respect to r, we obtain

∂M(r, t)

∂t+ v(r, t)

∂M(r, t)

∂r= 0. (80)

By using the expression of the gradient of the gravita-tional potential as given by Eq. (78), the Euler equation(76) becomes

∂v(r, t)

∂t+ v(r, t)

∂v(r, t)

∂r= −2u0

∂ρ(r, t)

∂r−

GM(r, t)

r2+

2

3Ω2(t)r. (81)

Equations (80) and (81) represent a system of two cou-pled nonlinear partial differential equations whose solu-tions for v and M fully determine the dynamical evo-lution of the collapsing Bose-Einstein Condensate dark

matter halo. This system of equations can be reduced toa single equation describing the dynamical evolution ofthe collapse. From Eq. (80) we obtain

v(r, t) = − M(r, t)

M ′(r, t). (82)

Substituting the above expression of the velocity intoEq. (81), and using the definition of the density in termsof the mass, we obtain the mass equation describingthe time evolution of the Bose-Einstein Condensate darkmatter halo as

− M(r, t)

M ′(r, t)+

2M(r, t)M ′(r, t)

M ′2(r, t)− M2(r, t)M ′′(r, t)

M ′3(r, t)=

u0

πr3M ′(r, t)− u0

2πr2M ′′(r, t)− GM(r, t)

r2+

2

3Ω2(t)r.(83)

A. The stationary solution

We consider first the case of the rotating station-ary (static) Bose-Einstein Condensate dark matter halos.The stationary case corresponds to ~v ≡ 0, M = M0(r),and Ω = constant, respectively. In this situation themass evolution equation Eq. (83) takes the form

d2M0(r)

dr2− 2

r

dM0(r)

dr+ a0M0(r) − b0r

3 = 0, (84)

where we have denoted

a0 =2πG

u0=

Gm3

~2a, (85)

and

b0 =4πΩ2

3u0=

2Ω2m3

3~2a, (86)

respectively. Eq. (84) has the general solution

M0(r) =

√

2

πa−3/40

[

(c1 −√a0c2r) sin (

√a0r)−

(√a0c1r + c2) cos (

√a0r)

]

+b0r

3

a0, (87)

where c1 and c2 are arbitrary constants of integration.The boundary condition

M0(0) = −√

2

πa−3/40 c2 = 0, (88)

gives c2=0, and therefore for the mass of the stationaryBose-Einstein Condensate dark matter halo we obtain

M0(r) =

√

2

πa−3/40 c1 [sin (

√a0r) −

√a0r cos (

√a0r)] +

b0r3

a0. (89)

14

For the density distribution of the rotating condensatewe obtain

ρ0(r) =3b04πa0

+4√a0c1 sin

(√a0r)

2√2π3/2r

(90)

The condition

ρ0(0) = ρc =a3/40 c1

2√2π3/2

+3b04πa0

, (91)

where ρc is the central density of the condensate, givesfor the integration constant c1 the expression

c1 =

√

π

2a−7/40 (4πa0ρc − 3b0) . (92)

Hence we finally obtain for the density and mass dis-tribution of the rotating Bose-Einstein Condensate darkmatter halos the expressions

ρ0(r) =Ω2

2πG+ ρc

(

1− Ω2

2πGρc

)

sin(√

a0r)

√a0r

, (93)

and

M0(r) =2Ω2

3Gr3 + 4πa

−3/20 ρc

(

1− Ω2

2πGρc

)

×

[sin (√a0r)−

√a0r cos (

√a0r)] , (94)

respectively. The radius R0 of the stationary condensateis determined from the condition ρ (R0) = 0, and it canbe obtained as a solution of the equation

sin (√a0R0) =

√a0R0

Ω2

2πρc

(

Ω2

2πGρc− 1

)−1

= −λ√a0R0,

(95)where we have denoted

λ =Ω2

2πGρc

(

1− Ω2

2πGρc

)−1

. (96)

By taking into account that the left hand side of theabove equation can be expanded in Taylor series as

sin (√a0R0) =

√a0R0−

1

6a3/20 R3

0+1

120a5/20 R5

0+O(

R6)

,

(97)we obtain for the determination of the radius of the rotat-ing condensate dark matter halo the algebraic equation

√a0R0

(

1 + λ− 1

6a0R

20 +

1

120a20R

40 + ...

)

= 0. (98)

In the first order of approximation we obtain for thestationary dark matter halo the approximate expression

R0 ≈√

6(1 + λ)

a0=

√

6

(1− Ω2/2πGρc)

~2a

Gm3. (99)

In the case of the static condensate with Ω ≡ 0, theradius RS of the dark matter halo is obtained exactly as

RS = π

√

~2a

Gm3= π

√a0 =

17.96×( a

10−3 fm

)1/2

×( m

meV

)−3/2

kpc.(100)

Then the radius of the rotating, stationary condensatedark matter halo can be also written in the form

R0 ≈√

6

π2 (1− Ω2/2πGρc)RS . (101)

The total mass of the rotating condensate dark matterhalo is obtained as

M0 (R0) =2Ω2

3GR3

0 −4π~2a

Gm3ρcR0

(

1− Ω2

2πGρc

)

×[

λ+√

1− λ2a0R20

]

. (102)

In the case of the static condensate√a0RS = π, and for

the total mass MS of the nonrotating dark matter halowe find

MS = 4πa−3/20 = 4π

(

~2a

Gm3

)3/2

ρc =4

πρcR

3S , (103)

or

MS = 3.17× 1010 ×( a

10−3 fm

)3/2

×( m

meV

)−9/2

×(

ρc10−24 g/cm3

)

M⊙. (104)

In the first order approximation the mass of the rotatingcondensate can be expressed in terms of the static radiusas

M0 (R0) ≈ 2Ω2

3GR3

0 −4√6

π2ρcR

3S

(

1− Ω2

2πGρc

)1/2

×

λ+

√

1−(

Ω2

2πGρc

)2(

1− Ω2

2πGρc

)−3

.

(105)

B. Collapsing Bose-Einstein Condensate darkmatter halos

Obtaining some analytical or semianalytical solutionsof the equations describing the time evolution of Bose-Einstein Condensate dark matter halos would allow toconstruct a clearer picture of the relationship betweenthe input physics and the behavior of the system, alsomaking possible to understand astrophysical effects thatare more difficult to understand with the extensive use of

15

numerical methods for solving the hydrodynamical evo-lution equations. One of the powerful mathematical ap-proaches for the study of the collapse processes in astro-physical phenomena is based on the idea of self-similarity,which consists in the rescaling of the radial coordinate ras r → ξ = r/α(t), where α(t) is a time only depen-dent function, and to assume that all the other physi-cal parameters can be expressed as functions of ξ andof some function of time, so that an arbitrary physicalquantity Φ(r, t) can be written as Φ(r, t) = β(t)φ(ξ). Aninteresting consequence of the self-similar (homologous)evolution is that the initial density and mass profiles ofthe collapsing systems do not change. From a mathe-matical point of view after introducing the appropriatelychosen self-similarity transformations, the system of non-linear partial differential equations (80) and (81) can bereduced in many cases to a system of ordinary nonlin-ear differential equations. Some astrophysically relevantself-similar solutions describe the gravitational collapseof isothermal spheres or of polytropic spheres. It wouldbe then interesting to consider self-similar solutions of thehydrodynamic equations describing the time evolution ofBose-Einstein Condensate Dark matter halos. Unfortu-nately, the hydrodynamic equations describing the evo-lution of a polytropic gas with equation of state p ∝ ρ2

do not admit self-similar or homologous solutions (for aproof of this results see Appendix A. The main reasonis that after introducing the self-similar variables in thestandard way, the explicit time dependence of the equa-tions cannot be eliminated for any choice of the timesimilarity functions.

1. The evolution equations

However, the dynamic evolution equation of the massof the Bose-Einstein Condensate dark matter halos, de-scribed by a polytropic equation of state with n = 1, andgiven by Eq. (83), does admit a separable, semi-analyticalsolution, which can be obtained under the assumptionthat the mass of the condensate can be represented as

M(r, t) = f(t)m(r), (106)

where f(t) andm(r) are functions depending on the time,and the radial coordinate r only. Then the velocity ofthe dark matter halo can be immediately obtained fromEq. (82) as

v(r, t) = − f(t)

f(t)

m(r)

m′(r). (107)

For the acceleration of the fluid we find

dv

dt=

(

− f(t)

f(t)+

f2(t)

f2(t)

)

m(r)

m′(r)+

f2(t)

f2(t)

m(r)

m′(r)

d

dr

m(r)

m′(r)=

(

− f(t)

f(t)+

f2(t)

f2(t)

)

m(r)

m′(r)+

1

2

f2(t)

f2(t)

d

dr

[

m(r)

m′(r)

]2

. (108)

Then the mass equation (83) can be written as

(

− f(t)

f(t)+

f2(t)

f2(t)

)

m(r)

m′(r)+

1

2

f2(t)

f2(t)

d

dr

[

m(r)

m′(r)

]2

=

f(t)

[

u0

πr3m′(r) − u0

2πr2m′′(r) − Gm(r)

r2

]

+2

3Ω2(t)r.

(109)

The explicit time dependence in Eq. (109) can be re-moved, and the variables can be separated, if the functionf(t) and the angular velocity Ω(t) satisfy the conditions

− f(t)

f(t)+

f2(t)

f2(t)= α0f(t), (110)

f2(t)

f2(t)=

4

β20

f(t), (111)

Ω2(t) = ω20f(t), (112)

where α0, β0 and ω0 are dimensional constants.Eq. (111) can be integrated immediately to give

f(t) =β20

(t0 ± t)2 , (113)

where t0 is an arbitrary constant of integration. ThenEq. (110) reduces to

− f(t)

f(t)+

f2(t)

f2(t)− α0f(t) = −2 + α0β

20

(t0 ± t)2 = 0, (114)

giving α0 = −2/β20 . Hence, the time dependence in the

mass evolution equation cancels out, and for the radialmass distribution of the time evolving dark matter halowe obtain the equation

[

u0β20

4πr2m′′(r) − u0β

20

2πr3m′(r) +

Gβ20m(r)

2r2

]

− 1

3ω20β

20r =

m(r)

m′(r)− d

dr

[

m(r)

m′(r)

]2

. (115)

In the above equation β0 has the physical unit of second,while ω0 has the units 1/s. The function f(t) is dimen-sionless. Hence once the solution of Eq. (115) is known,the evolution of the physical parameters of the collaps-ing/expanding Bose-Einstein dark matter halos are givenby

M(r, t) =β20

(t0 ± t)2m(r), ρ(r, t) =

β20

4πr2 (t0 ± t)2m

′(r),

v(r, t) =2

(t0 ± t)

m(r)

m′(r),Ω(t) = ± β0ω0

t0 ± t. (116)

16

We rescale the radial coordinate r, the mass m(r), thedensity ρ(r, t) and the velocity v(r, t) according to thetransformations

r =Rs

πη, m(r) = MSm0(η),

where RS =√

πu0/2G and MS = (4/π)R3S ρc are the

radius and the mass of the static condensate, while ηand m0(η) are the dimensionless radial coordinate andthe dimensionless mass function, respectively. We denoteγ20 = ω2

0/6π2Gρc. Moreover, we also fix in the following

the constant β20 as

β20 =

1

4π2Gρc. (117)

Hence for the physical parameters of the collaps-ing/expanding Bose-Einstein Condensate dark matterhalo we obtain the expressions

M(η, t) =MS

4π2Gρc

m0(η)

(t0 ± t)2 , ρ(η, t) =

1

4πG

1

(t± t0)2 θ (η) ,

v(η, t) =2

(t± t0)

RS

πV (η),Ω(t) = ± ω0

√

4π2Gρc

1

t0 ± t.

(118)

The dimensionless spatial density distribution of thedensity θ (η) and velocity V (η) of the Bose-Einstein Con-densate dark matter halo can be obtained as

θ(η) =1

η2m′

0(η), V (η) =m0(η)

m′(η). (119)

Then, in the new dimensionless variables, Eq. (115),describing the mass profile of the evolving dark matterhalo, takes the form

m′′0 (η)−

2

ηm′

0(η) +m0(η) − γ20η

3 =

2η2

m0(η)

m′0(η)

− d

dη

[

m0(η)

m′0(η)

]2

, (120)

or, equivalently,[

1− 4η2m2

0(η)

m′ 30 (η)

]

m′′0(η) −

2

ηm′

0(η) +

[

1 +2η2

m′0(η)

]

m0(η)− γ20η

3 = 0. (121)

For the central density of the condensate at the be-ginning of the contracting/expanding phase we obtainρ(0, 0) = ρc = θ(0)/4πGt20, which gives θ(0) = θ0 =4πGρct

20.

C. The first order approximation

In the zeroth order approximation, we can neglect theright hand side in Eq. (120), which thus takes the form

m′′0(η)−

2

ηm′

0(η) +m0(η)− γ20η

3 = 0, (122)

and has the general solution

m(0)0 (η) = γ2

0η3 −

√

2

π

[

(C2η − C1) sin(η) +

(C1η + C2) cos(η)

]

, (123)

where C1 and C2 are arbitrary constants of integration.

The boundary condition m(0)0 (0) = 0 gives C2 = 0. For

the dimensionless density we obtain

θ(0)(η) =

√

2

πC1

sin(η)

η+ 3γ2

0 . (124)

The boundary condition θ(0)(0) = θ0 gives C1 =√

π/2(

θ0 − 3γ20

)

. Hence we obtain the zeroth order so-lution of Eq. (120) as

m(0)0 (η) = γ2

0η3 +

(

θ0 − 3γ20

)

[sin η − η cos η] . (125)

The density is given by

θ(0)(η) =

(

θ0 − 3γ20

)

sin η

η+ 3γ2

0 . (126)

To obtain the solution of Eq. (120) in the first order ofapproximation we substitute in its right hand side m(η)

by m(0)0 (η), thus obtaining

h(η) = 2η2

m0(η)

m′0(η)

− d

dη

[

m0(η)

m′0(η)

]2

=2η3

9+

(

2γ20

9− 2

27

)

η5 +

(

−1197γ40 + 663γ2

0 − 88)

η7

4725+O

(

η8)

.

(127)

Hence in this approximation Eq. (120) becomes

m′′0 (η)−

2

ηm′

0(η) +m0(η) =

(

γ20 +

2

9

)

η3 +

(

2γ20

9− 2

27

)

η5 +

(

−1197γ40 + 663γ2

0 − 88)

η7

4725,(128)

with the general solution given by

m(1)0 (η) = −7

(

9576γ40 − 5139γ2

0 + 574)

η3

945+

14(

3γ20 − 1

) (

798γ20 − 151

)

η5

4725+

(

−1197γ40 + 663γ2

0 − 88)

η7

4725−

√

2

π[(c4η − c3) sin η + (c3η + c4) cos η] ,(129)

where c3 and c4 are arbitrary constants of integration.

The boundary condition m(1)0 (0) = 0 gives C4 = 0, while

17

the density profile of the Bose-Einstein Condensate darkmatter halo is obtained as

θ(1)(η) =

√

2

πc3sin η

η+

(

−1197γ40 + 663γ2

0 − 88)

η4

675+

2

135

(

2394γ40 − 1251γ2

0 + 151)

η2 +

1

45

(

−9576γ40 + 5139γ2

0 − 574)

. (130)

By imposing the boundary condition θ(1)(0) = 1 we ob-

tain C3 =√

π/2(

9576γ40 − 5139γ2

0 + 619)

/45. Hence forthe spatial mass and density profiles of the dynamicallyevolving Bose-Einstein Condensate dark matter halos weobtain, in the first order approximation, the expressions

m(1)0 (η) =

(

−1197γ40 + 663γ2

0 − 88)

η7

4725+

2

675

(

2394γ40 − 1251γ2

0 + 151)

η5 +

1

135

(

−9576γ40 + 5139γ2

0 − 574)

η3 +

1

45

(

9576γ40 − 5139γ2

0 + 619)

sin η −1

45

(

9576γ40 − 5139γ2

0 + 619)

η cos η,(131)

and

θ(1)(η) =1

675

(

−1197γ40 + 663γ2

0 − 88)

η4 +

2

135

(

2394γ40 − 1251γ2

0 + 151)

η2 +(

9576γ40 − 5139γ2

0 + 619)

sin η

45η+

1

45

(

−9576γ40 + 5139γ2

0 − 574)

, (132)

respectively. In the first order approximation the spa-tial velocity profile of the evolving Bose-Einstein Con-densate dark matter halo can be obtained as V (1)(η) =

m(1)0 (η)/dm

(1)0 (η)/dη.

The above results simplify considerably in the caseof the nonrotating condensate dark matter halos, withγ20 = 0. Hence, for a radially expanding/contracting

Bose-Einstein Condensate confined by its own gravita-tional field we obtain

m(1)0 (η) = −88η7

4725+302η5

675− 574η3

135+619

45(sin η − η cos η) ,

(133)

θ(1)(η) =619 sinη

45η− 2

675

(

44η4 − 755η2 + 4305)

, (134)

V (1)(η) =1

η+

88η6 − 2730η4 + 30660η2 + 64995 cosη − 60270

7 (88η5 − 1510η3 + 8610η − 9285 sinη). (135)

In this approximation the mass and density are finiteat the origin η = 0, but the velocity diverges at the centerof the contracting condensate dark matter halo.

D. Exact numerical profiles

Eq. (121) can be reformulated mathematically as a firstorder dynamical system given by

dm0

dη= u, (136)

du

dη=

2

η

u

1− 4η2m20/u

3−(

1 + 2η2/u)

m0

1− 4η2m20/u

3+

γ20η

3

1− 4η2m20/u

3. (137)

The system of equations (136) and (137) must beintegrated with the initial conditions m0(0) = 0 andu(0) = 0, respectively. As for the dimensionless den-sity θ and velocity V (η) of the Bose-Einstein Condensatedark matter they can be represented as

θ(η) =1

η2u, V (η) =

m0

u. (138)

0.0 0.5 1.0 1.5

2.5

3.0

3.5

η

θ(η)

FIG. 1: Variation of the dimensionless density profile θ(η) asa function of the dimensional radial coordinate η of a collaps-ing rotating Bose-Einstein Condensate dark matter halo fordifferent values of the parameter γ0: static case with γ0 = 0,(solid curve), and dynamical evolution with γ0 = 0 (dottedcurve), γ0 = 0.25 (short dashed curve), γ0 = 0.50 (dashedcurve), γ0 = 0.75 (long dashed curve), and γ0 = 0.90 (longdashed curve), respectively. To numerically integrate the sys-tem of equations (136) and (137) we have used the initial con-ditions m0 (η0) = (4/π) η3

0 and u (η0) = m′

0 (η0) = (12/π) η2

0 ,with η0 = 10−3.

The variations of the dimensionless spatial density,mass and velocity profiles of the contracting/expandingBose-Einstein Condensate dark matter halos are repre-sented in Figs. 1-3, respectively. To numerically integrate

18

0.0 0.5 1.0 1.5

0

1

2

3

4

5

η

m0(η)

FIG. 2: Variation of the dimensionless mass profile m0(η) asa function of the dimensional radial coordinate η of a collaps-ing rotating Bose-Einstein Condensate dark matter halo fordifferent values of the parameter γ0: static case with γ0 = 0,(solid curve), and dynamical evolution with γ0 = 0 (dottedcurve), γ0 = 0.25 (short dashed curve), γ0 = 0.50 (dashedcurve), γ0 = 0.75 (long dashed curve), and γ0 = 0.90 (longdashed curve), respectively. To numerically integrate the sys-tem of equations (136) and (137) we have used the initial con-ditions m0 (η0) = (4/π) η3

0 and u (η0) = m′

0 (η0) = (12/π) η2

0 ,with η0 = 10−3.

0.0 0.5 1.0 1.5

0.0

0.2

0.4

0

η

V

(η)

FIG. 3: Variation of the dimensionless velocity profile V (η)as a function of the dimensional radial coordinate η of a col-lapsing rotating Bose-Einstein Condensate dark matter halofor different values of the parameter γ0 during its dynam-ical evolution for γ0 = 0 (solid curve), γ0 = 0.25 (dottedcurve), γ0 = 0.50 (short dashed curve), γ0 = 0.75 (dashedcurve), and γ0 = 0.90 (long dashed curve), respectively. Tonumerically integrate the system of equations (136) and (137)we have used the initial conditions m0 (η0) = (4/π) η3

0 andu (η0) = m′

0 (η0) = (12/π) η2

0 , with η0 = 10−3.

the system of equations (136) and (137) we have usedthe initial conditions m0 (η0) = (4/π) η30 and m′

0 (η0) =(12/π) η20 , with η0 = 10−3. The spatial density profileof the dark matter halo during its dynamical evolutionis represented in Fig. 1. The static solution in the ab-sence of rotation is also represented, as the lower curvein the figure. In all cases the density profiles are de-

scribed by monotonically decreasing functions of the di-mensionless radius η. The effects of the rotation of thehalo can be clearly seen, and they lead to a significantimpact on the density profiles. Near the origin, for η inthe range 0 < η < 0.25, the density profiles are basicallyindistinguishable, but for larger values of η the effectsof the rotation modify the overall density distribution.The impact of the rotation on the spatial mass distri-bution of the collapsing/expanding dark matter halos ispresented in Fig. 2. The mass profiles are characterizedby linearly increasing functions of η. For η in the range0 < η < 1.1, the mass distribution of the rotating collaps-ing/expanding condensate dark matter halo essentiallycoincides with the static mass distribution. The effectsof the rotation become important for large values of theradius, once we are approaching the lower density regionsat the outer boundary of the halo. The rotation can leadto a significant increase in the total mass of the halo, theincrease being of the order of 20% for γ0 = 0.9.The spatial velocity profiles of the dynamically evolv-

ing condensate dark matter halos are presented in Fig. 3.Similarly to the mass profile, the spatial distribution ofthe velocity is a monotonically increasing function of η.For η in the range 0 < η < 1.1, the velocity profilesare (almost) identical, and the effects of rotation are ex-tremely small. For small rotation velocities the velocitycan be well approximated by a linear function of η, sothat V (η) ∝ η. However, for larger values of the di-mensionless radial coordinate, the effects of the rotationbecome important at large distances from the halo cen-ter, and the velocity - radial coordinate relations is notlinear anymore.

V. DISCUSSIONS AND FINAL REMARKS

In the present paper we have investigated some of thepossible physical and astrophysical consequences of theinstabilities in a Bose-Einstein Condensate dark matterhalo. We have considered two distinct, but interrelatedtopics, the Jeans stability of condensate dark matterclouds, and the dynamics of the gravitational collapsethat follows once the real size of the gravitationally con-fined system exceeds the Jeans scales.The Bose-Einstein Condensates are generally described

by the Gross-Pitaevskii equation, which gives the groundstate of a quantum system of identical bosons, and whichis obtained by using the Hartree-Fock approximation andthe pseudopotential interaction model. In the presentapproach we assume that dark matter is a Bose-EinsteinCondensate of a gas of bosons, which are in the samequantum state, and thus can be described by the samewavefunction. Hence, we interpret the galactic dark mat-ter halos as huge quantum systems, whose properties canbe described by a single wave function. To simplify ourformalism we also adopt the assumption that dark mat-ter is at zero temperature. Of course the presence of theexcitations and of nonzero temperature effects may have

19

important effects on the properties of condensate darkmatter.

An important mathematical and physical property ofthe Gross-Pitaevskii equation is that, similarly to thestandard Schrodinger equation, it admits a hydrodynam-ical representation, easily obtainable after representingthe wave function in the Madelung variables. It turnsout that in the hydrodynamic representation the Bose-Einstein Condensates can be described as a perfect fluid,and its properties are characterized by the local densityand local velocity only. The fluid satisfies a continuityand an Euler type equations, which contains the contri-butions of the quantum pressure p = u0ρ

2, generated bythe self-interaction of the field, and of the quantum po-tential VQ, both of these terms being essentially quantummechanical in their origin. The velocity ~v of the quantumfluid is related to the phase S of the wave function by therelation ~v = (~/m)∇S, and the flow of the condensatefluid is rotationless, as long as S contains no singulari-ties, as, for example, in a vortex. In the present approachwe have also allowed the presence of vortices in the darkmatter halo. The simplest method to create vortices ina condensate is through their rotation. Hence, underthe natural assumption that Bose-Einstein Condensatedark matter halos are rotating, one must also take intoaccount the existence of vortices in the system. A Bose-Einstein Condensate dark matter halo can rotate onlydue to the existence of quantized vortex lines. Whenthe rotation frequency Ω of the halo exceeds a criticalvalue Ωc, vortex nucleation occurs. In the present paperwe have adopted the simple relation (22) between theangular velocity of the rotating condensate dark matterhalos, and their vortex number density, which indicatesthat the quicker the dark matter halo rotates, the higherthe number of vortices. On the other hand in rotatinggravitationally trapped Bose-Einstein Condensate darkmatter halos the nucleation of vortices may also be a re-sult of the instabilities of collective excitations.

In the present manuscript we have used the term tur-bulence in the sense that it is common in the physicsof quantum fluids, meaning the presence in the fluid ofquantized vortices [188, 189]. In this nomenclature tur-bulence is more related to the quantization of vorticesthan to the lack of viscosity of superfluids. On the otherhand the term turbulence may also be interpreted as de-scribing a phase of temporally and spatially disorderedfluid motion, characterized by a large number of degreesof freedom interacting essentially nonlinearly [189]. Thenonlinear interaction is usually a consequence of the pres-ence of the term (~v · ∇)~v in the Euler equation.

The hydrodynamical description of the rotating con-densate dark matter halos, confined by their own grav-itational fields, opens the possibility of the study of thegravitational perturbations in galactic and extragalacticsystems. By assuming some standard initial conditions,the linearization of the hydrodynamic equations describ-ing the flow of the dark matter clouds leads to the dis-persion relation (33), which is exact in the sense that it

also includes both the effects of the rotation and of thequantum force. In the limit when the rotation and thequantum force effects are neglected, Eq. (33) reduces tothe standard form of the Jeans frequency,

ω2 = v2s~k2 − 4πGρ0, (139)

and to the Jeans radius RJ =(

√

π/Gρ0

)

vs. The Jeans

radius essentially depends on the speed of sound in thegiven astrophysical medium, and it is a measure of thelinear dimensions of the condensations which will formin the medium on account of the gravitational insta-bility. For matter obeying the ideal gas equation ofstate p = (kBT/mµ), where T is the temperature ofthe gas, µ is the mean molecular weight, and kB isthe Boltzmann constant, we have vs =

√

kBT/mµ, and

RJ =√

πkBT/Gρ0mµ. For an ideal gas the Jeans lengthis proportional to the temperature of the system, and in-versely proportional to the density. On the other handfor a Bose-Einstein Condensate in the same approxima-tion we obtain RJ =

√

4π2~2a/Gm3, and expression thatdepends only on the fundamental physical characteris-tics of the dark matter halo (particle mass and scatter-ing length), and on the Planck constant. Hence, for aBose-Einstein Condensate at zero temperature the Jeansradius has an universal expression, independent on themacroscopic properties of the medium.In the opposite limit of zero speed of sound (or negligi-

ble quantum pressure), the dispersion relation takes theform

ω2 =~2

4m2~k4 − 4πGρ0

(

1 +~Ω20

πGρ0

)

, (140)

giving a Jeans wave number given by

∣

∣

∣

~k∣

∣

∣

J=

16πGρ0m2(

1 + ~Ω20/πGρ0

)

~2

1/4

, (141)

and a Jeans radius of the order of

RJ =

π3~2

Gρ0m2(

1 + ~Ω20/πGρ0

)

1/4

. (142)

In the absence of rotation, Eqs. (141) and (142) givethe Jeans wave number and the Jeans radius correspond-ing to the Schrodinger-Poisson model, described by thesystem of equations [190–194]

i~∂Ψ(~r, t)

∂t= − ~

2

2m∆Ψ(~r, t) +mVg (~r, t)Ψ (~r, t) , (143)

∆Vg (~r, t) = 4πGm2 |Ψ(~r, t)|2 , (144)

where Vg (~r, t) is the gravitational potential. The es-sential difference between the Schrodinger-Poisson and

20

Gross-Pitaevskii-Poisson models of dark matter is thatin the former the effect of the particle self-interaction isneglected, which implies that in the Schrodinger-Poissonmodel there is no quantum pressure term p, and the speedof sound in the dark matter halo is zero. By assuminga zero rotational velocity, for the Jeans radius and Jeansmass of the dark matter halo in the Schrodinger-Poissonmodel we obtain the relations

RJ =

[

π3~2

Gρ0m2

]1/4

= 3.57× 1012 ×( m

meV

)−1/2

×(

ρ010−24 g/cm3

)−1/4

cm, (145)

and

MJ =

(

28/3π10/3~2

34/3Gm2

)3/4

ρ1/40 =

9.532× 10−20 ×( m

meV

)−3/2(

ρ010−24 g/cm3

)1/4

M⊙,

(146)

respectively. In the Schrodinger-Poisson model all thephysical and astrophysical properties of the dark matterhalos are fixed by a single parameter, the mass of the darkmatter particle. In order to obtain realistic astrophysicalresults applicable to the galactic or extra-galactic scalesvery small particle masses are required, a problem theBose-Einstein Condensate dark matter model does nothave.We have also discussed the effects of the rotation on the

condensate dark matter halos. In realistic astrophysical

situations the condition ~Ω2/πGρ0 >> 1 may be satisfied.In this case the Jeans wave number is proportional tothe angular velocity of the halo, while the Jeans radius

and Jeans mass are inversely proportional to∣

∣

∣

~Ω0

∣

∣

∣, and∣

∣

∣

~Ω0

∣

∣

∣

3

, respectively. For large angular velocities the Jeans

radius is very small, of the order of 10−2 kpc, while thecorresponding Jeans mass is of the order of 90M⊙. Theseradius and mass values may correspond to stellar massblack holes, or other types of small (in an astrophysicalsense) compact objects. In the case of ordinary baryonicmatter the condition of instability for waves propagatingperpendicularly to the axis of rotation is k2Jv

2s < 4πGρ0−

4Ω2 [159].One constraint on the ratio and scattering length of the

dark matter particle can be obtained from the condition2GMJ/c

2RJ < 1, requiring that dark matter halos can-not become black holes. This condition is essentially gen-eral relativistic, but we may assume that it is also valid inNewtonian mechanics. By adopting for the Jeans radiusand mass the expressions (47) and (50), respectively, weobtain the constraint

a

m3<

3c2

32π3~2ρ0= 2.449×1096

(

ρ010−24 g/cm3

)−1

cm/g3.

(147)

Once the numerical values of the critical parameters ofthe condensate dark matter halos are reached, an insta-bility develops in the system. One can interpret, from asimple physical point of view, the collapse of the Bose-Einstein Condensate dark matter halos as follows. Oncethe number of particles N in the galactic halo becomessufficiently high, so that N > Nc, where Nc is a criticalparticle number, the attractive inter-particle energy ex-ceeds the quantum pressure, and the dark matter halobegins to collapse. During the collapsing stage, the den-sity of dark matter particles increases in the vicinity ofthe galactic center.

To study the dynamical evolution of the halo oneneed to solve the full system of hydrodynamic equationsdescribing the time and space evolution of the Bose-Einstein Condensate. In the Thomas-Fermi approxima-tion the basic equations describing the dynamics of thecondensate coincide with the continuity and Euler equa-tions of classical hydrodynamics. The effect of the ro-tation was also included by means of a term, averagedover the angle θ, proportional to the radial coordinate,and with a time-dependent angular frequency. In order topreserve spherical symmetry we assume that the rotationof the condensate is slow. The effect of the gravitationalfield is taken into account via the Poisson equation, whichrelates the gravitational potential of the confined conden-sate to its density, which is essentially quantum in its na-ture. After integrating the Poisson equation, the evolu-tion equations can be reduced to a system of two couplednonlinear partial differential equations, which can subse-quently be reduced to a single nonlinear second order par-tial differential equations for the mass of the condensateM(r, t). For many hydrodynamic flow models in gravita-tional fields, with matter obeying a polytropic equation ofstate the equations of motion admit semi-analytical self-similar or homologously collapsing solutions [195–199].

When these solutions are valid scale radial profiles ofthermodynamic variables remain time invariant. The ho-mologous solutions are stable against linear nonsphericalperturbations, a result which can explains the remarkablestability of the galactic dark matter halos. However, inthe case of index n = 1 polytropes no self-similar solu-tion does exist, and the solutions of the equations of mo-tion cannot be written in the form Φ(r, t) = f(t)g (r/t)[195–199]. This means that during their evolution thephysical quantities describing Bose-Einstein Condensatedark matter halos are not scale and time invariant.However an exact semi-analytical solution of the evo-lution equations of the Bose-Einstein Condensates self-confined by the gravitational field can be obtained byassuming the separability of the mass variable in the forM(r, t) = f(t)m(r). The function f(t) has the analytical

form f(t) = (t0 ± t)−2

/4π2Gρc, where the minus signcorresponds to the collapse of the condensate, while theplus sign describes the expansion of the condensate darkmatter halo. As for the mass function m(r), time evolu-tion generates a universal condensate dark matter profile,which in the absence of rotation just slightly modifies the

21

static mass and density profiles. However, once the rota-tion is taken into account, the mass and density profilesare significantly modified as functions of the angular ve-locity.The obtained solution describes both an expanding

and a collapsing phase. The expanding phase corre-sponds to the choice of plus sign in the function f(t), so

that f(t) = (t0 + t)−2

/4π2Gρc. The Bose-Einstein Con-densate disperses in the space, with the local values ofthe physical parameters decreasing in time. In the largetime limit the density as well as the radial and angularvelocities tend to zero, and the Bose-Einstein Condensatereaches a vacuum state. The minus sign in the functionf(t) describes the collapse of the condensate. Both solu-tions are singular, for the expanding case the singularityoccurs at the initial moment t = 0 (by a rescaling of thetime variable one can remove the constant t0 from theresults, t → t+ t0), while in the case of the gravitationalcollapse the singularity is reached at the finite time t0. Inorder to obtain a rough estimate of t0, we assume that themass distribution m(r) during the collapsing phase doesnot differ much from the stationary one. By series ex-panding Eq. (94) we obtain m(r) ≈ M0(r) ≈ (4π/3)ρcr

3,an expression which allows us to estimate the velocityof the collapsing Bose=Einstein Condensate dark matterhalos as

v(r, t) ≈ 2

t0 − t

r

3. (148)

At the beginning of the gravitational collapse at t = 0,the extension of the condensate cloud is given by its Jeansradius RJ , which we approximate by Eq. (47). More-over, we assume that the initial velocity of the collapsingboundary can be written as v (RJ , 0) = αc, where α is aconstant. Then for the collapse time t9 of the condensatehalo we obtain

t0 ≈ 2RJ

3αc=

23.30

α×1011×

( m

meV

)−3/2 ( a

10−3 fm

)1/2

s.

(149)

An alternative estimate of the collapse time can beobtained by assuming that at the beginning of the col-lapse the total mass of the condensate dark matter haloM (RJ , 0) = m (RJ ) is given by the Jeans mass (50).Then we obtain for the collapse time the simple expres-sion t0 ≈ 1/2π

√Gρc.

It is interesting to note that, at least in the first orderof approximation, the velocity of the condensate darkmatter halo has the mathematical form of the cosmolog-ical Hubble law, v(t) = H(t)r, with the Hubble functiongiven by H(t) = 2/ (t0 ± t). Moreover, as shown by thenumerical simulations presented in Fig. 3, for small an-gular velocities the simple linear proportionality betweenthe dimensionless velocity V and the radial dimension-less coordinate η is valid during the entire phase of thecollapsing/expanding evolution.

A large number of cosmological and astrophysical ob-servations, including the behavior of the galactic rotationcurves, or the virial mass discrepancy in galaxy clusterspoints towards the possibility of the existence of darkmatter in the Universe. If dark matter is present in theform of bosonic particles, then the possibility that it is inthe form of a Bose-Einstein Condensate is strongly sup-ported by the laws of quantum physics that show that aphase transition to a condensate state must occur, oncethe temperature reaches its critical value. A major ad-justment in our interpretation of the basic principles ofastrophysics and cosmology will be required if further ob-servations would confirm this hypothesis. In the presentpapers we have introduced some basic theoretical toolsthat could help in the analysis and interpretation of thelarge scale structure formation in the presence of Bose-Einstein Condensate dark matter.

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Appendix A: No self-similar contraction/expansionof Bose-Einstein Condensate dark matter halos

Eqs. (80) and (81) are invariant under the time reversaloperations t → −t, v → −v, and ρ → ρ, M → M ,respectively. Hence, without any loss of generality, onecan restrict the analysis to the range 0 ≤ t < ∞ of thetime variable. In order to investigate the possibility ofthe existence of a self-similar solution that describes thecollapse of Bose-Einstein Condensate dark matter halos,we introduce first a similarity variable ξ, and assume thateach physical quantity scales according to the followingrules,

r = α(t)ξ, v(r, t) = β(t)u (ξ) ,

ρ(r, t) = γ(t)θ (ξ) ,M(r, t) = η(t)m (ξ) , (A1)

where α(t), β(t), γ(t) and η(t) are continuous functionsof the time variable t only.Then we obtain first

dm(ξ)

dξ= 4π

γ(t)α3(t)

η(t)ξ2θ(ξ). (A2)

In the self-similar variables the continuity equation

25

(80) gives

m(ξ) =η(t)

η(t)α(t)[α(t)ξ − β(t)u(ξ)]

dm(ξ)

dξ=

4πγ(t)

η(t)α2(t)ξ2θ(ξ) [α(t)ξ − β(t)u(ξ)] ,(A3)

while the Euler equation (81 takes the form

u(ξ) +β(t)

α(t)β(t)[β(t)u(ξ) − α(t)ξ]

du(ξ)

dξ=

−2u0γ(t)

α(t)β(t)

dθ(ξ)

dξ− Gη(t)

α2(t)β(t)

m(ξ)

ξ2+

2

3Ω2(t)

α(t)

β(t)ξ.

(A4)

The existence of a similarity solution requires that thecondition

α(t) = β(t), (A5)

must be satisfied by the functions α(t) and β(t). Theexplicit time dependence in the evolution equations ofthe condensate dark matter halo could be removed if theconditions

γ(t)α3(t)

η(t)= c1,

γ(t)

η(t)α2(t)α(t) = c2,

α2(t)

α(t)α(t)= c3,

(A6)

2u0γ(t)

α(t)α(t)= c4,

Gη(t)

α2(t)α(t)= c5,

2

3Ω2(t)

α(t)

α(t)= c6,

(A7)

are also satisfied, where ci, i = 1, ..., 6 are constants. Thethird relation in Eqs. (A6) determines the function α(t)as a solution of the second order differential equation

α(t) =α2(t)

c3α(t), (A8)

with a particular solution given by

α(t) = C1tn, (A9)

where C1 is an arbitrary constant of integration, and n =c3/ (c3 − 1). The first two of the equations (A7) give

γ(t) =c42u0

α(t)α(t), η(t) =c5Gα2(t)α(t). (A10)

The substitution of the above relations into the first ofEqs. (A6) gives α2(t) = constant, a result that obviouslycontradicts Eq. (A9). Hence the explicit time dependencein the hydrodynamic equations describing the dynamicsof a polytropic fluid with equation of state p ∼ ργ , γ = 2,cannot be removed by means of the similarity transfor-mations (A1), and therefore there are no self-similar solu-tions describing the evolution of Bose-Einstein Conden-sate dark matter halos. However, for polytropic fluidswith γ 6= 2 the hydrodynamic type evolution equationsalways admit self-similar solutions.